Systems and Methods for Dual-Carrier Modulation Encoding and Decoding

ABSTRACT

Systems and methods for dual-carrier modulation (DCM) encoding and decoding for communication systems. Some embodiments comprise a DCM encoder for applying a pre-transmission function to at least one 16-QAM input symbol and mapping resulting transformed symbols onto at least one larger constellation prior to transmission. Some embodiments joint decode, by a DCM decoder, a predetermined number of received data elements and compute a set of log-likelihood ratio (LLR) values for at least eight bits from a resulting at least one transformed symbol.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation-in-part and claims priority to U.S.application Ser. No. 12/099,308 which claims priority to U.S.provisional patent application Ser. No. 60/912,487 for “Dual-CarrierModulation (DCM) Encoder-Decoder for Higher Data Rate Modes of WiMediaPHY”, hereby incorporated herein by reference.

BACKGROUND

As devices become increasingly mobile and interoperable, networks may bemore than the customary established grouping of devices. Instead, or insome cases in addition, devices join and leave networks on an ad-hocbasis. Such devices may join an existing network, or may form atemporary network for a limited duration or for a limited purpose. Anexample of such networks might be a personal area network (PAN). A PANis a network used for communication among computer devices (includingmobile devices such as laptops, mobile telephones, game consoles,digital cameras, and personal digital assistants) which are proximatelyclose to one person. Any of the devices may or may not belong to theperson in question. The reach of a PAN is typically a few tens ofmeters. PANs can be used for communication among the personal devicesthemselves (ad-hoc communication), or for connecting to a higher levelnetwork and/or the Internet (infrastructure communication). Personalarea networks may be wired, e.g., a universal serial bus (USB) and/orIEEE 1394 interface or wireless. The latter communicates via networkingtechnologies consistent with the protocol standards propounded by theInfrared Data Association (IrDA), the Bluetooth Special Interest Group(Bluetooth), the WiMedia Alliance's ultra wideband (UWB), or the like.

Among recently emerging communication technologies—especially thoseneeding high data transfer rates—various ultra-wideband (UWB)technologies are gaining support and acceptance. UWB technologies areutilized for wireless transmission of video, audio or other highbandwidth data between various devices. Generally, UWB is utilized forshort-range radio communications—typically data relay between deviceswithin approximately 10 meters—although longer-range applications may bedeveloped. A conventional UWB transmitter generally operates over a verywide spectrum of frequencies, several GHz in bandwidth. UWB may bedefined as radio technology that has either: 1) spectrum that occupiesbandwidth greater than 20% of its center frequency; or, as is it is morecommonly understood, 2) a bandwidth≧500 MHz.

Next generation networks, such as those standardized by the WiMediaAlliance, Inc., increase the range, speed, and reliability of wirelessdata networks. One implementation of next generation networks utilizesultra-wideband (UWB) wireless technology, specifically a MultiBandorthogonal frequency-division multiplexing (OFDM) physical layer (PHY)radio along with a sophisticated medium access control (MAC) layer thatcan deliver data rates up to 480 megabits per second (Mbps).

The WiMedia UWB common radio platform enables high-speed (up to 480Mbps), low power consumption data transfers in a wireless personal areanetwork (WPAN). The WiMedia UWB common radio platform incorporates MAClayer and PHY layer specifications based on MultiBand OFDM (MB-OFDM).WiMedia UWB is optimized for the personal computer (PC), consumerelectronics (CE), mobile device and automotive market segments. ECMA-368and ECMA-369 are international ISO-based specifications for the WiMediaUWB common radio platform. Additional information may be found in U.S.patent application Ser. No. 11/099,317, entitled “Versatile System forDual Carrier Transformation in Orthogonal Frequency DivisionMultiplexing”, and U.S. patent application Ser. No. 11/551,980, entitled“Dual-Carrier Modulation Decoder”, which are incorporated herein byreference.

Increasing demand for more powerful and convenient data and informationcommunication has resulted in a number of advancements, particularly inwireless communication technologies. Despite the advancements, however,significant improvement in data transfer rates is sought.

BRIEF DESCRIPTION OF THE DRAWINGS

For a detailed description of exemplary embodiments of the invention,reference will be made to the accompanying drawings in which:

FIG. 1 is a block diagram of a DCM system, according to embodiments;

FIG. 2 illustrates a sixteen point constellation, which may be used toadvantage by embodiments;

FIG. 3 illustrates an exemplary embodiment of encoding;

FIG. 4A illustrates reduced complexity log-likelihood ratio (LLR)equation for LLR(b_(4k)), according to embodiments;

FIG. 4B illustrates reduced complexity LLR equation for LLR(b_(4k+1)),according to embodiments;

FIG. 4C illustrates reduced complexity LLR equation for LLR(b_(4(k+p))),according to embodiments;

FIG. 4D illustrates reduced complexity LLR equation forLLR(b_(4(k+p)+1)), according to embodiments;

FIG. 4E illustrates reduced complexity LLR equation for LLR(b_(4k+2)),according to embodiments;

FIG. 4F illustrates reduced complexity LLR equation for LLR(b_(4k+3)),according to embodiments;

FIG. 4G illustrates reduced complexity LLR equation forLLR(b_(4(k+p)+2)), according to embodiments;

FIG. 4H illustrates reduced complexity LLR equation forLLR(b_(4(k+p)+3)), according to embodiments;

FIG. 41 illustrates reduced complexity LLR equation for LLR(b_(4k)) whennoise variance across tones is not uniform, according to embodiments;

FIG. 4J illustrates reduced complexity LLR equation for LLR(b_(4k+1))when noise variance across tones is not uniform, according toembodiments;

FIG. 4K illustrates reduced complexity LLR equation for LLR(b_(4(k+p)))when noise variance across tones is not uniform, according toembodiments;

FIG. 4L illustrates reduced complexity LLR equation forLLR(b_(4(k+p)+1)), when noise variance across tones is not uniform,according to embodiments;

FIG. 4M illustrates reduced complexity LLR equation for LLR(b_(4k+2))when noise variance across tones is not uniform, according toembodiments;

FIG. 4N illustrates reduced complexity LLR equation for LLR(b_(4k+3))when noise variance across tones is not uniform, according toembodiments;

FIG. 4O illustrates reduced complexity LLR equation forLLR(b_(4(k+p)+2)) when noise variance across tones is not uniform,according to embodiments;

FIG. 4P illustrates reduced complexity LLR equation forLLR(b_(4(k+p)+3)) when noise variance across tones is not uniform,according to embodiments;

FIG. 5 illustrates an exemplary embodiment of decoding;

FIG. 6 illustrates an exemplary embodiment of decoding usingapproximation;

FIG. 7 illustrates an exemplary general-purpose computer system suitablefor implementing the several embodiments of the disclosure; and

FIG. 8 illustrates an exemplary MAC, PHY, and MAC-PHY interface suitablefor implementing the several embodiments of the disclosure.

NOTATION AND NOMENCLATURE

Certain terms are used throughout the following description and claimsto refer to particular system components. As one skilled in the art willappreciate, computer companies may refer to a component by differentnames. This document does not intend to distinguish between componentsthat differ in name but not function. In the following discussion and inthe claims, the terms “including” and “comprising” are used in anopen-ended fashion, and thus should be interpreted to mean “including,but not limited to . . . .” Also, the term “couple” or “couples” isintended to mean either an indirect or direct electrical connection.Thus, if a first device couples to a second device, that connection maybe through a direct electrical connection, or through an indirectelectrical connection via other devices and connections. The term“system” refers to a collection of two or more hardware and/or softwarecomponents, and may be used to refer to an electronic device or devicesor a sub-system thereof. Further, the term “software” includes anyexecutable code capable of running on a processor, regardless of themedia used to store the software. Thus, code stored in non-volatilememory, and sometimes referred to as “embedded firmware,” is includedwithin the definition of software.

DETAILED DESCRIPTION

It should be understood at the outset that although exemplaryimplementations of embodiments of the disclosure are illustrated below,embodiments may be implemented using any number of techniques, whethercurrently known or in existence. This disclosure should in no way belimited to the exemplary implementations, drawings, and techniquesillustrated below, including the exemplary design and implementationillustrated and described herein, but may be modified within the scopeof the appended claims along with their full scope of equivalents.

In light of the foregoing background, embodiments provide systems andmethods for dual carrier modulation (DCM) to accommodate 16-QAM inputconstellations, which systems and methods are particularly useful forexploiting frequency diversity for data rates higher than 480 Mbps.

Although embodiments will be described for the sake of simplicity withrespect to wireless communication systems, it should be appreciated thatembodiments are not so limited, and can be employed in a variety ofcommunication systems.

Despite the readily apparent advantages of using high data transferrates, e.g., 640 Mbps, 800 Mbps, 960 Mbps, and above, no form ofredundancy other than convolutional coding is used (neitherfrequency-domain spreading nor time-domain spreading), and therefore,prior to the present invention, there was no way to exploit the fullfrequency diversity of the channel in such a system. Of course, for datatransfer rates at or below 480 Mbps, the WiMedia Alliance physical (PHY)layer version 1.0 (WiMedia Alliance/Multi-band OFDM Alliance PhysicalLayer Specification Version 1.x) uses dual-carrier modulation (DCM) toexploit frequency diversity at the cost of slightly increasedcomplexity; see also for example, and without limitation, U.S. patentapplication Ser. No. 11/099,317 for “Versatile System for Dual CarrierTransformation in Orthogonal Frequency Divisional Multiplexing” and U.S.patent application Ser. No. 11/551,980, entitled “Dual-CarrierModulation Decoder”, hereby incorporated by reference herein.

One technique for increasing physical data rate is to use a 16-QAMconstellation instead of quadrature phase-shift keying (QPSK); examplesof such a technique may be found in U.S. patent application Ser. No.11/115,816 for “Multi-band OFDM High Data Rate Extensions,” herebyincorporated by reference herein. Table 1 summarizes potential datarates when using a 16-QAM constellation.

TABLE 1 PSDU rate-dependent parameters Coding Data Rate Rate CodedBits/6 Info Bits/6 (Mb/s) Modulation (R) FDS TDS OFDM symbols OFDMsymbols 106.7 16-QAM ⅓ YES YES 600 200 160 16-QAM ½ YES YES 600 300213.3 16-QAM ⅓ NO YES 1200 400 320 16-QAM ½ NO YES 1200 600 400 16-QAM ⅝NO YES 1200 750 640 16-QAM ½ NO NO 2400 1200 800 16-QAM ⅝ NO NO 24001500 960 16-QAM ¾ NO NO 2400 1800

Some of the present embodiments are hereafter illustratively describedin conjunction with the design and operation of an ultra-wideband (UWB)communications system utilizing an Orthogonal Frequency DivisionMultiplexing (OFDM) scheme. Certain aspects of the present disclosureare further detailed in relation to design and operation of a MultiBandOFDM (MB-OFDM) UWB communications system. Although described in relationto such constructs and operations, the teachings and embodimentsdisclosed herein may be beneficially implemented with any datatransmission or communication systems or protocols (e.g., IEEE 802.11(a)and IEEE 802.11(n)), depending upon the specific needs or requirementsof such systems. ECMA International has published WiMedia Alliancestandard ECMA-368 entitled, “High Rate Ultra Wideband PHY and MACStandard”, and ECMA-369 entitled, “MAC-PHY Interface for ECMA-368”,which are hereby incorporated herein by reference as if reproduced infull, and which can be utilized in conjunction with the presentembodiments.

Embodiments of OFDM-based wireless communication systems utilize apre-transmission conversion function to convert a data signal from thefrequency domain into the time domain for over-the-air transmission overa wireless channel. During transmission over the wireless channel, somedegree of signal noise and possibly interference is added to the timedomain data signal. As the time domain signal is received, apost-transmission conversion function is utilized to convert the signalback into the frequency domain, for subsequent signal processing orcommunication. Often, such pre-transmission and post-transmissionconversion functions take the form of Inverse Fast Fourier Transforms(IFFTs) and Fast Fourier Transforms (FFTs), respectively.

Within the context of an OFDM-based UWB system, a pre-transmission IFFTcommonly has 128 points (or tones) at the nominal baud rate. Dependingupon the type of communications system, or specific design orperformance requirements, however, an IFFT may have any desired orrequired number of tones. In some embodiments, one hundred of thosetones are used as data carriers, twelve are pilot carriers (i.e., carrydata known to receiver that it uses to ensure coherent detection), tenare guard carriers, and six are null tones. The ten guard carriers maybe configured to serve a number of concurrent or independent functions.For example, some portion of the guard tones may be configured toimprove signal-to-noise ratios (SNRs), by loading those guard carrierswith critical data (e.g., unreliable data) for redundant transmission.Some portion of the guard tones may be configured (e.g., leftunutilized) as frequency guard bands, to prevent interference to or fromadjacent frequency bands; see also for example, and without limitation,U.S. patent application Ser. No. 11/021,053 for “Mapping Data Tones ontoGuard Tones for a Multi-band OFDM system”, hereby incorporated byreference herein. Of the six null tones, one typically occupies themiddle of the available signal spectrum, and the others may beselectively configured or designated to conform to a desired spectralmask (e.g., UWB, 802.11, 802.16).

Within a MB-OFDM system, data tones may be loaded with quadratureamplitude modulation (QAM) data. For a high-throughput MB-OFDM system,there are a number of techniques that may be used to manipulate ortailor system data rates. In addition to code puncturing, techniquessuch as frequency domain spreading and time domain spreading may beemployed to divide data transmission to a desired data rate. Frequencyspreading and time spreading are two techniques which introduceredundancy into the transmission of data. However, as noted above, withemerging wireless technology, frequency spreading and time spreading arenot available techniques at very high data rates, e.g., at or above 320Mbps. By applying DCM to such systems, a different form of redundancy isincluded when frequency domain spreading or time domain spreadingtechniques are not available due to the high data rates. Unfortunately,known techniques for applying DCM to QPSK are not possible for datarates above 480 Mbps. Therefore, one of the innovative features of thepresent disclosure is the ability to compensate for the lack offrequency spreading or time spreading by providing embodiments whichenable DCM to accommodate 16-QAM input constellations, which embodimentsare particularly useful for exploiting frequency diversity for datarates higher than 480 Mbps.

The disclosure, in some embodiments, provides embodiments forimplementing a dual carrier modulation (DCM) encoder which concurrentlymaps eight separate input bits; four of which are mapped onto one 16-QAMconstellation to form a tone set or symbol, while the remaining four ofthe eight input bits are mapped onto a second 16-QAM constellation toform a second tone set or symbol. Encoder embodiments group the two16-QAM symbols into a vector and apply a conversion or transformationfunction to the vector to together produce two transformed symbols, thefirst of which is mapped onto a first 256-point constellation and thesecond of which is mapped onto a second 256-point constellation. Itshould be understood that the mapping used to create the first 256-pointconstellation and the second 256-point constellation may, in someembodiments, be dissimilar. This dissimilar arrangement is intended torefer to the placement of a symbol to a first location within the first256-point constellation and the placement of the same symbol to a secondlocation within the second 256-point constellation. Encoder embodimentsmap these transformed symbols onto two separate tones. The first toneand second tone are then transmitted with a predetermined frequencyspacing inserted in between the transmission of the first tone and thesecond tone. One of the innovative features of transmitting a pair ofsymbols selected from two possibly different 256-point constellationpair is the ability at the receiver to maximize available informationthrough the combination of the first symbols and the second symbols. Insome situations, if a tone is lost or faded, then the symbol transmittedon that tone may be recovered by using information from the symboltransmitted on the other tone pair. By spreading the data over two toneswith a large spacing between them, the probability that all elements ofthe tone pair will be degraded is greatly reduced. While embodimentsdiscussed herein use a 256 point constellation, it is expresslyunderstood that any number of points could be used to create aconstellation. It is further expressly understood that while the samenumber of points are used in embodiments discussed herein,constellations containing a dissimilar number of points may be usedconsistent with the disclosed embodiments.

Another embodiment for implementing a dual carrier modulation (DCM)encoder is to map the eight input bits onto a first 256-pointconstellation directly to produce a first transformed symbol and map thesame eight input bits onto a second 256-point constellation directly toproduce a second transformed symbol.

The present disclosure, in some embodiments, provides systems andmethods for implementing a dual carrier modulation (DCM) decoder withjoint decoding. Joint decoding accepts a stream of data which has beenseparated into a first tone and a second tone distanced by somepredetermined frequency spacing from the first tone. The first tone andsecond tone may be referred to collectively as a single tone pair. Thephrase joint decoding (sometimes referred to herein as a joint decoder)is intended to refer to the apparatus, system or method by which a DCMdecoder concurrently uses two separate tones to decode the tone pair andrecover reliability information about the eight input bits. As notedabove, if one of the bits within one of the tones is lost or degraded,it can be identified or recovered by embodiments of a DCM decoder.

FIG. 1 is a block diagram 10 of a system using DCM 16. Thecommunications system comprises transmitter 12 and receiver 36.Transmitter 12 comprises encoder 14, which in turn comprises DCM 16 andIFFT 18. Receiver 36 comprises decoder 38, which in turn comprises FFT40 and LLR calculator 46. It should be appreciated that either or bothtransmitter 12 and receiver 36 may comprise additional functional blocksfor further processing (e.g., noise variance estimation, channelestimation, FEC decoder, synchronization, scaling of channel outputs,etc.) The input to encoder 14 is a first symbol selected from a 16-QAMconstellation 20 bearing data elements (e.g., 4 bits) and a secondsymbol selected from a 16-QAM constellation 22 bearing data elements(e.g., 4 more bits). An example of a 16-QAM constellation is illustratedin FIG. 2. Each 16-QAM constellation symbol represents four dataelements. A data element may include, but is not limited to, a singlebit of data. DCM 16 performs a transform on first symbol 20 togetherwith second symbol 22 to create first transformed symbol 24 and secondtransformed symbol 26. Each of transformed symbols 24 and 26 bear thedata from all eight data elements corresponding to symbols 20 and 22. Analternative description of exemplary embodiment of DCM 16 is to take the4 bits corresponding to first symbol 20 and take the 4 bitscorresponding to second symbol 22 and map the concatenated 8 bits onto afirst 256-point constellation to produce first transformed symbol 24 andmap the same concatenated 8 bits onto a second 256-point constellationto produce second transformed symbol 26.

Returning to FIG. 1, DCM 16 maps first transformed symbol 24 onto afirst tone and second transformed symbol 26 onto a second tone. It iscontemplated that a frequency spacing 28 may, in some embodiments, existbetween the first tone 24 and second tone 26. In some embodiments, thisspacing may be maximized in order to maximize the frequency diversity.The examples discussed herein use a fifty-tone spacing, however, it isexpressly understood that any number of tone spacings could be used, andthat more or less than a fifty tone spacing could be used consistentwith this disclosure. First tone (or symbol) 24 and second tone (orsymbol) 26 are passed into IFFT 18. IFFT 18 performs an inversefast-Fourier transform to produce a time-domain signal 34. In someembodiments, signal 34 is further processed (shown for the sake ofdiscussion as 34′) before arriving at decoder 38. Signal 34 (or 34′) isforwarded to FFT 40 where the signal is passed through a fast-Fouriertransformation, resulting in recovery of third tone 42 which may besimilar to first tone 24 and fourth tone 44 which may be similar to thesecond tone 26 from the signal. Decoder 38 groups tones 42 and 44 topass to LLR calculator 46 so that these two tones (symbols) may bejointly decoded to produce reliability information. Here alog-likelihood ratio is performed jointly on the symbols 42 and 44before the combined resulting set of LLRs is forwarded for furtherprocessing or analysis within receiver 36. More specifics with respectto encoding and decoding are provided below.

Focusing first on embodiments implementing DCM encoding, as noted above,dual-carrier modulation was added to the WiMedia physical layer version1.0 specification in order to exploit channel diversity at the higherdata rates with QPSK constellation inputs, where no additional forms ofspreading other than possibly the convolutional code were available.However, embodiments of the present disclosure enable DCM to be extendedto 16-QAM input constellations.

FIG. 3 is a flowchart of an exemplary embodiment of encoding. In thisexample embodiment, bits b₀, b₁, b₂, b₃ are mapped onto a first input16-QAM constellation and bits b₄, b₅, b₆, b₇ are mapped onto a secondinput 16-QAM constellation to produce symbol₁ and symbol₂, respectively(block 310). Symbol₁ and symbol₂, are grouped into a vector and passedthrough an orthonormal matrix to together produce transformed symbol₁and transformed_symbol₂ which are symbols belonging to a first and asecond 256-point constellation, respectively (block 320).Transformed_symbol₁ and transformed_symbol₂ are then mapped onto twoseparate tones (block 330). In some embodiments, the tones may befurther processed prior to being transmitted.

The mapping between the normalized 16-QAM symbol on the k^(th) tone,s_(k), and the DCM symbol on the k^(th) tone, d_(k), is illustrated asequation (1):

D_(k)=TS_(k)  (1)

where D_(k)=[d_(k) d_(k+p)]^(T), S_(k)=[s_(k) s_(k+p)]^(T), p is anon-zero integer. For example, but not by way of limitation, the WiMediaphysical layer version 1.0 specification uses a total of 100 data tones;therefore, to ensure maximum exploitation of the channel diversity avalue of 50 is selected for p. It is expressly understood that if thenumber of data tones should change, the value of p should also beaccordingly changed.

The mixing matrix from equation (1) is illustrated as equation (2):

$\begin{matrix}{T = {\frac{1}{\sqrt{17}}\begin{bmatrix}4 & 1 \\1 & {- 4}\end{bmatrix}}} & (2)\end{matrix}$

In at least one embodiment, each DCM symbol, d_(k), is selected from a256-point constellation using a unique mapping of the eight bits ofinformation contained in S_(k), where T=T^(T). Note that other possiblemixing matrices exist, which can be summarized by the following matrixillustrated as equation (2′):

$\begin{matrix}{T = {\frac{1}{\sqrt{17}}\begin{bmatrix}{\pm 4} & {\pm 1} \\{\pm 1} & {\pm 4}\end{bmatrix}}} & \left( 2^{\prime} \right)\end{matrix}$

It should be appreciated that the signs, in at least some embodiments,are selected to ensure a one-to-one mapping between the two input 16-QAMconstellations and the two 256-point constellations. Equation (1) iswritten explicitly as:

$\begin{bmatrix}d_{k} \\d_{k + p}\end{bmatrix} = \begin{bmatrix}{{T_{1,1} \cdot s_{k}} + {T_{1,2} \cdot s_{k + p}}} \\{{T_{2,1} \cdot s_{k}} + {T_{2,2} \cdot s_{k + p}}}\end{bmatrix}$

If the matrix T is any of the matrices represented by

${\frac{1}{\sqrt{17}}\begin{bmatrix}{\pm 4} & {\pm 1} \\{\pm 1} & {\pm 4}\end{bmatrix}},$

then

$d_{i} \in {\left\{ {\frac{+ 3}{\sqrt{17}},\frac{+ 1}{\sqrt{17}},\frac{- 1}{\sqrt{17}},\frac{- 3}{\sqrt{17}}} \right\} + {j{\left\{ {\frac{+ 3}{\sqrt{17}},\frac{+ 1}{\sqrt{17}},\frac{- 1}{\sqrt{17}},\frac{- 3}{\sqrt{17}}} \right\}.}}}$

Similarly, if the rows of the matrix T are reversed, the same set ofvalues is obtained

$d_{i} \in {\left\{ {\frac{+ 3}{\sqrt{17}},\frac{+ 1}{\sqrt{17}},\frac{- 1}{\sqrt{17}},\frac{- 3}{\sqrt{17}}} \right\} + {j{\left\{ {\frac{+ 3}{\sqrt{17}},\frac{+ 1}{\sqrt{17}},\frac{- 1}{\sqrt{17}},\frac{- 3}{\sqrt{17}}} \right\}.}}}$

So the matrices represented by

$T = {\frac{1}{\sqrt{17}}\begin{bmatrix}{\pm 1} & {\pm 4} \\{\pm 4} & {\pm 1}\end{bmatrix}}$

are effectively equivalent to those represented by

$T = {{\frac{1}{\sqrt{17}}\begin{bmatrix}{\pm 4} & {\pm 1} \\{\pm 1} & {\pm 4}\end{bmatrix}}.}$

In 16-QAM, s_(k), can be written into its real and imaginary componentsas:

s _(k) =Re(s _(k))+j*Im(s _(k)).

When writing out the real and imaginary components explicitly, equation(1) is equivalent to:

D _(k) =T·Re(S _(k))+T·Im(S _(k))

Since the T matrix operates independently on the real and imaginaryparts, the same T matrix need not be used. So another way equation thatis effectively the same is:

D _(k) =T ₁ ·Re(S _(k))+T ₂ ·Im(S _(k))

where T₁ and T₂ can each be different variations of the T matrix definedalready defined. When T₁≠T₂, the elements of D_(k) still belong to thesame set:

$d_{i} \in {\left\{ {\frac{+ 3}{\sqrt{17}},\frac{+ 1}{\sqrt{17}},\frac{- 1}{\sqrt{17}},\frac{- 3}{\sqrt{17}}} \right\} + {j{\left\{ {\frac{+ 3}{\sqrt{17}},\frac{+ 1}{\sqrt{17}},\frac{- 1}{\sqrt{17}},\frac{- 3}{\sqrt{17}}} \right\}.}}}$

Consider now embodiments implementing DCM decoding. In an exemplaryembodiment, the maximum-likelihood DCM decoder approach is used. In thisembodiment, after FFT 40, the received signal 34′ r_(k) for the k^(th)tone can be written as equation (3)

r _(k) =h _(k) d _(k) +n _(k)  (3)

where h_(k) is the channel coefficient for the k^(th) tone, and n_(k) isa complex white Gaussian random variable with varianceE[n_(k)n_(k)*]=σ_(k) ². It should be appreciated in the presentdiscussion, that it is assumed that the noise variance is the sameacross the tones, i.e., σ_(k) ²=σ² for all k.

The received vector R_(k)=[r_(k) r_(k+p)]^(T), which represents thereceived signal 34′, in at least some embodiments, for the k^(th) and(k+p)^(th) tones, can be written in matrix notation as shown in equation(4):

R _(k) =H _(k) D _(k) +N _(k)  (4)

where, in some embodiments, N_(k)=[n_(k) n_(k+p)]^(T). One example of arepresentation for H_(k) is:

$\begin{matrix}{H_{k} = {\begin{bmatrix}h_{k} & 0 \\0 & h_{k + p}\end{bmatrix}.}} & (5)\end{matrix}$

The output used in this embodiment from the frequency-domain equalizer(FEQ), Y_(k), can be written as:

Y _(k) =H _(k) *R _(k) =|H _(k)|² D _(k) +H _(k) *N _(k) =|H _(k)|² TS_(k) +H _(k) *N _(k)  (6)

where, in at least one embodiment,

$\begin{matrix}{{H_{k}}^{2} = {\begin{bmatrix}{h_{k}}^{2} & 0 \\0 & {h_{k + p}}^{2}\end{bmatrix}.}} & (7)\end{matrix}$

In some embodiments, the interleaved and coded bits [b_(4k) b_(4k+1)b_(4k+2) b_(4k+3)], —where the subscript for the first bit is the valueequal to four (4) times k, the subscript for the second bit is four (4)times k plus one (1), the subscript for the third bit is four (4) timesk plus two (2), and the subscript for the fourth bit is four (4) times kplus three (3)—are mapped onto the normalized 16-QAM constellationsymbol s_(k) and [b_(4(k+p)) b_(4(k+p)+1) b_(4(k+p)+2)b_(4(k+p)+3)]—where the subscript for the first bit is the value equalto four (4) times (k+p), the subscript for the second bit is four (4)times (k+p) plus one (1), the subscript for the third bit is four (4)times (k+p) plus two (2), and the subscript for the fourth bit is four(4) times (k+p) plus three (3)—are mapped on to the normalized 16-QAMconstellation symbol s_(k+p). It is expressly understood that althoughthe notation implies four consecutive bits are mapped onto the symbols,in actuality the four bits mapped onto each symbol can be selected ormapped in any order from the interleaved and coded bits, for example,and not by way of limitation, [b_(4k) b_(4(k+p)+2) b_(4(k+p)) b_(4k+1)].Because, in some embodiments, the underlying 16-QAM constellation isGray-coded and the mixing matrix T is real, the equation for Y_(k) canbe separated into its real and imaginary parts, and each portion can beindependently optimized:

Re(Y _(k))=|H _(k)|² TRe(S _(k))+Re(H _(k) *N _(k))  (8)

Im(Y _(k))=|H _(k)|² TIm(S _(k))+Im(H _(k) *N _(k))  (9)

Note that this effectively subdivides MIMO detection with complex inputsinto two separate MIMO detection portions with real inputs. It isexpressly understood that whereas the complexity with complex inputsgrows as |

|², where |

| is the number of elements in the input constellation, the complexitywith real inputs grows as 2|

|.

Given the previous equations, the log-likelihood ratio (LLR) for b_(4k),b_(4k+1), b_(4(k+p)), b_(4(k+p)+1) (bits that map onto the real axis)and b_(4k+2), b_(4k+3), b_(4(k+p)+2), b_(4(k+p)+3) (bits that map ontothe imaginary axis) can be determined. In some embodiments, the LLR isused to determine the probability that a single discrete point, bit ortone is accurately received.

The LLR for b_(4k) is given by:

$\begin{matrix}\begin{matrix}{{{LLR}\left( b_{4k} \right)} = {\log \left\lbrack \frac{\sum\limits_{b_{{4k} + 1}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}}^{\;}\; {\Pr \begin{pmatrix}{{{Y_{k}b_{4k}} = 0},b_{{4k} + 1},} \\{b_{4{({k + p})}},b_{{4{({k + p})}} + 1}}\end{pmatrix}}}{\sum\limits_{b_{{4k} + 1}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}}^{\;}\; {\Pr \begin{pmatrix}{{{Y_{k}b_{4k}} = 1},b_{{4k} + 1},} \\{b_{4{({k + p})}},b_{{4{({k + p})}} + 1}}\end{pmatrix}}} \right\rbrack}} \\{= {\log \left\lbrack \frac{\begin{matrix}{\sum\limits_{{{b_{{4k} + 1}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}},{b_{4k} = 0}}^{\;}\exp} \\\left\lbrack {{- \left( {\alpha_{k} - {{H_{k}}^{- 2}{TS}_{R}}} \right)^{T}}\frac{2}{\sigma^{2}}{H_{k}}^{2}\left( {\alpha_{k} - {{H_{k}}^{2}{TS}_{R}}} \right)} \right\rbrack\end{matrix}}{\begin{matrix}{\sum\limits_{{{b_{{4k} + 1}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}},{b_{4k} = 1}}^{\;}\exp} \\\left\lbrack {{- \left( {\alpha_{k} - {{H_{k}}^{2}{TS}_{R}}} \right)^{T}}\frac{2}{\sigma^{2}}{H_{k}}^{- 2}\left( {\alpha_{k} - {{H_{k}}^{2}{TS}_{R}}} \right)} \right\rbrack\end{matrix}} \right\rbrack}}\end{matrix} & (11)\end{matrix}$

where α_(k)=[α₁ α₂]^(T)=[Re(y_(k)) Re(y_(k+o))]^(T) and S_(R)=Re(S_(k)).By expanding equation (11) and eliminating the terms that do not dependon S_(R), the previous equation can be re-written as:

$\begin{matrix}{{{LLR}\left( b_{4k} \right)} = {\log\left\lbrack \frac{\begin{matrix}{\sum\limits_{{{b_{{4k} + 1}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}},{b_{4k} = 0}}^{\;}\exp} \\\left\lbrack {\frac{2}{\sigma^{2}}S_{R}^{T}{T\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack\end{matrix}}{\begin{matrix}{\sum\limits_{{{b_{{4k} + 1}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}},{b_{4k} = 1}}^{\;}\exp} \\\left\lbrack {\frac{2}{\sigma^{2}}S_{R}^{T}{T\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack\end{matrix}} \right\rbrack}} & (12)\end{matrix}$

Note that the summation over b_(4k+1), b_(4(k+p)), b_(4(k+p)+1) withb_(4k)=0 implies using the following vectors for S_(R) in the summationin equation (12):

$S_{R} = {\frac{1}{\sqrt{10}}{\left\{ {\begin{bmatrix}{- 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 3}\end{bmatrix}} \right\}.}}$

Further, the summation over b_(4k+1), b_(4(k+p)), b_(4(k+p)+1) withb_(4k=1) implies using the following vectors for S_(R) in the summationin equation (12):

$S_{R} = {\frac{1}{\sqrt{10}}{\left\{ {\begin{bmatrix}{+ 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 3}\end{bmatrix},\begin{bmatrix}{+ 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 3} \\{- 3}\end{bmatrix}} \right\}.}}$

Similarly, the LLR expression for b_(4k+1) can be written as:

$\begin{matrix}{{{LLR}\left( b_{{4k} + 1} \right)} = {\log\left\lbrack \frac{\begin{matrix}{\sum\limits_{{{b_{4k}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}},{b_{{4k} + 1} = 0}}^{\;}\exp} \\\left\lbrack {\frac{2}{\sigma^{2}}S_{R}^{T}{T\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack\end{matrix}}{\begin{matrix}{\sum\limits_{{{b_{4k}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}},{b_{{4k} + 1} = 1}}^{\;}\exp} \\\left\lbrack {\frac{2}{\sigma^{2}}S_{R}^{T}{T\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack\end{matrix}} \right\rbrack}} & (13)\end{matrix}$

Note that the summation over b_(4k), b_(4(k+p)), b_(4(k+p)+1) withb_(4k+1)=0 implies using the following vectors for S_(R) in thesummation in equation (13):

$S_{R} = {\frac{1}{\sqrt{10}}{\left\{ {\begin{bmatrix}{- 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 3}\end{bmatrix},\begin{bmatrix}{+ 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 3} \\{- 3}\end{bmatrix}} \right\}.}}$

Further, the summation over b_(4k), b_(4(k+p)), b_(4(k+p)+1) withb_(4k+1)=1 implies using the following vectors for S_(R) in thesummation in equation (13):

$S_{R} = {\frac{1}{\sqrt{10}}{\left\{ {\begin{bmatrix}{- 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 3}\end{bmatrix}} \right\}.}}$

Similarly, the LLR expression for b_(4(k+p)) can be written as:

$\begin{matrix}{{{LLR}\left( b_{4{({k + p})}} \right)} = {\log\left\lbrack \frac{\begin{matrix}{\sum\limits_{{{b_{4k}b_{{4k} + 1}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}},{b_{4{({k + p})}} = 0}}^{\;}\exp} \\\left\lbrack {\frac{2}{\sigma^{2}}S_{R}^{T}{T\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack\end{matrix}}{\begin{matrix}{\sum\limits_{{{b_{4k}b_{{4k} + 1}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}},{b_{4{({k + p})}} = 1}}^{\;}\exp} \\\left\lbrack {\frac{2}{\sigma^{2}}S_{R}^{T}{T\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack\end{matrix}} \right\rbrack}} & (14)\end{matrix}$

Note that the summation over b_(4k), b_(4k+1), b_(4(k+p)+1) withb_(4(k+p))=0 implies using the following vectors for S_(R) in thesummation in equation (14):

$S_{R} = {\frac{1}{\sqrt{10}}{\left\{ {\begin{bmatrix}{+ 3} \\{- 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 3}\end{bmatrix},\begin{bmatrix}{+ 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 1}\end{bmatrix}} \right\}.}}$

Further, the summation over b_(4k), b_(4k+1), b_(4(k+p)+1) withb_(4(k+p))=1 implies using the following vectors for S_(R) in thesummation in equation (14):

$S_{R} = {\frac{1}{\sqrt{10}}{\left\{ {\begin{bmatrix}{+ 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 3}\end{bmatrix}} \right\}.}}$

Similarly, the LLR expression for b_(4(k+p)+1) can be written as:

$\begin{matrix}{{{LLR}\left( b_{{4{({k + p})}} + 1} \right)} = {\log \left\lbrack \frac{\begin{matrix}{\sum\limits_{{{b_{4k}b_{{4k} + 1}b_{4{({k + p})}}} \in {\{{0,1}\}}},{b_{{4{({k + p})}} + 1} = 0}}^{\;}\exp} \\\left\lbrack {\frac{2}{\sigma^{2}}S_{R}^{T}{T\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack\end{matrix}}{\begin{matrix}{\sum\limits_{{{b_{4k}b_{{4k} + 1}b_{4{({k + p})}}} \in {\{{0,1}\}}},{b_{{4{({k + p})}} + 1} = 1}}^{\;}\exp} \\\left\lbrack {\frac{2}{\sigma^{2}}S_{R}^{T}{T\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack\end{matrix}} \right\rbrack}} & (15)\end{matrix}$

Note that the summation over b_(4k), b_(4k+1), b_(4(k+p)) withb_(4(k+p)+1)=0 implies using the following vectors for S_(R) in thesummation in equation (15):

$S_{R} = {\frac{1}{\sqrt{10}}{\left\{ {\begin{bmatrix}{+ 3} \\{- 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 3}\end{bmatrix},\begin{bmatrix}{+ 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 3}\end{bmatrix}} \right\}.}}$

Further, the summation over b_(4k), b_(4k+1), b_(4(k+p)) withb_(4(k+p)+1)=1 implies using the following vectors for S_(R) in thesummation in equation (15):

$S_{R} = {\frac{1}{\sqrt{10}}{\left\{ {\begin{bmatrix}{+ 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 1}\end{bmatrix}} \right\}.}}$

Similar expressions can also be derived for the bits that correspond tothe imaginary axis. For example, the LLR expression for b_(4k+2) can bewritten as:

$\begin{matrix}{{{LLR}\left( b_{{4k} + 2} \right)} = {\log \left\lbrack \frac{\begin{matrix}{\sum\limits_{{{b_{{4k} + 3}b_{{4{({k + p})}} + 2}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}},{b_{{4k} + 2} = 0}}^{\;}\exp} \\\left\lbrack {\frac{2}{\sigma^{2}}S_{I}^{T}{T\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack\end{matrix}}{\begin{matrix}{\sum\limits_{{{b_{{4k} + 3}b_{{4{({k + p})}} + 2}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}},{b_{{4k} + 2} = 1}}^{\;}\exp} \\\left\lbrack {\frac{2}{\sigma^{2}}S_{I}^{T}{T\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack\end{matrix}} \right\rbrack}} & (16)\end{matrix}$

where β_(k)=[β₁ β₂]^(T)=[Im(y_(k)) Im(y_(k+p))]^(T) and S_(I)=Im(S_(k)).Note that the summation over b_(4k+3), b_(4(k+p)+2), b_(4(k+p)+3) withb_(4k+2)=0 implies using the following vectors for S_(I) in thesummation in equation (16):

$S_{I} = {\frac{1}{\sqrt{10}}{\begin{Bmatrix}{\begin{bmatrix}{- 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 3}\end{bmatrix},} \\{\begin{bmatrix}{- 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 3}\end{bmatrix}}\end{Bmatrix}.}}$

Further, the summation over b_(4k+3), b_(4(k+p)+2), b_(4(k+p)+3) withb_(4k+2)=1 implies using the following vectors for S_(I) in thesummation in equation (16):

$S_{I} = {\frac{1}{\sqrt{10}}{\begin{Bmatrix}{\begin{bmatrix}{+ 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 3}\end{bmatrix},} \\{\begin{bmatrix}{+ 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 3} \\{- 3}\end{bmatrix}}\end{Bmatrix}.}}$

Similarly, the LLR expression for b_(4k+3) can be written as:

$\begin{matrix}{{{LLR}\left( b_{{4k} + 3} \right)} = {\log\left\lbrack \frac{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4{({k + p})}} + 2}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}}, \\ b_{{4k} + 3} = 0}}{\exp \left\lbrack {\frac{2}{\sigma^{2}}S_{I}^{T}{T\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}}{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4{({k + p})}} + 2}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}}, \\ b_{{4k} + 3} = 1}}{\exp \left\lbrack {\frac{2}{\sigma^{2}}S_{I}^{T}{T\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}} \right\rbrack}} & (17)\end{matrix}$

Note that the summation over b_(4k+2), b_(4(k+p)+2), b_(4(k+p)+3) withb_(4k+3)=0 implies using the following vectors for S_(I) in thesummation in equation (17):

$S_{I} = {\frac{1}{\sqrt{10}}{\begin{Bmatrix}{\begin{bmatrix}{- 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 3}\end{bmatrix},} \\{\begin{bmatrix}{+ 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 3} \\{- 3}\end{bmatrix}}\end{Bmatrix}.}}$

Further, the summation over b_(4k+2), b_(4(k+p)+2), b_(4(k+p)+3) withb_(4k+3)=1 implies using the following vectors for S_(I) in thesummation in equation (17):

$S_{I} = {\frac{1}{\sqrt{10}}{\begin{Bmatrix}{\begin{bmatrix}{- 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 3}\end{bmatrix},} \\{\begin{bmatrix}{+ 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 3}\end{bmatrix}}\end{Bmatrix}.}}$

Similarly, the LLR expression for b_(4(k+p)+2) can be written as:

$\begin{matrix}{{{LLR}\left( b_{{4{({k + p})}} + 2} \right)} = {\log \left\lbrack \frac{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4k} + 3}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}}, \\ b_{{4{({k + p})}} + 2} = 0}}{\exp \left\lbrack {\frac{2}{\sigma^{2}}S_{I}^{T}{T\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}}{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4k} + 3}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}}, \\ b_{{4{({k + p})}} + 2} = 1}}{\exp \left\lbrack {\frac{2}{\sigma^{2}}S_{I}^{T}{T\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}} \right\rbrack}} & (18)\end{matrix}$

Note that the summation over b_(4k+2), b_(4k+3), b_(4(k+p)+3) withb_(4(k+p)+2)=0 implies using the following vectors for S_(I) in thesummation in equation (18):

$S_{I} = {\frac{1}{\sqrt{10}}{\begin{Bmatrix}{\begin{bmatrix}{+ 3} \\{- 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 3}\end{bmatrix},} \\{\begin{bmatrix}{+ 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 1}\end{bmatrix}}\end{Bmatrix}.}}$

Further, the summation over b_(4k+2), b_(4k+3), b_(4(k+p)+3) withb_(4(k+p)+2)=1 implies using the following vectors for S_(I) in thesummation in equation (18):

$S_{I} = {\frac{1}{\sqrt{10}}{\begin{Bmatrix}{\begin{bmatrix}{+ 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 1}\end{bmatrix},} \\{\begin{bmatrix}{+ 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 3}\end{bmatrix}}\end{Bmatrix}.}}$

Similarly, the LLR expression for b_(4(k+p)+3) can be written as:

$\begin{matrix}{{{LLR}\left( b_{{4{({k + p})}} + 3} \right)} = {\log\left\lbrack \frac{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4k} + 3}b_{{4{({k + p})}} + 2}} \in {\{{0,1}\}}}, \\ b_{{4{({k + p})}} + 3} = 0}}{\exp \left\lbrack {\frac{2}{\sigma^{2}}S_{I}^{T}{T\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}}{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4k} + 3}b_{{4{({k + p})}} + 2}} \in {\{{0,1}\}}}, \\ b_{{4{({k + p})}} + 3} = 1}}{\exp \left\lbrack {\frac{2}{\sigma^{2}}S_{I}^{T}{T\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}} \right\rbrack}} & (19)\end{matrix}$

Note that the summation over b_(4k+2), b_(4k+3), b_(4(k+p)+2) withb_(4(k+p)+3)=0 implies using the following vectors for S_(I) in thesummation in equation (19):

$S_{I} = {\frac{1}{\sqrt{10}}{\begin{Bmatrix}{\begin{bmatrix}{+ 3} \\{- 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 3}\end{bmatrix},} \\{\begin{bmatrix}{+ 3} \\{+ 3}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 3}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 3}\end{bmatrix}}\end{Bmatrix}.}}$

Further, the summation over b_(4k+2), b_(4k+3), b_(4(k+p)+2) withb_(4(k+p)+3)=1 implies using the following vectors for S_(I) in thesummation in equation (19):

$S_{I} = {\frac{1}{\sqrt{10}}{\begin{Bmatrix}{\begin{bmatrix}{+ 3} \\{- 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{- 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{- 1}\end{bmatrix},} \\{\begin{bmatrix}{+ 3} \\{+ 1}\end{bmatrix},\begin{bmatrix}{+ 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 1} \\{+ 1}\end{bmatrix},\begin{bmatrix}{- 3} \\{+ 1}\end{bmatrix}}\end{Bmatrix}.}}$

It is expressly understood that it is possible to simplify the equationseven further by multiplying the matrices by the vectors. It isunderstood that this is an exact approach to the calculation of the LLR.However, approximations may be used in order to reduce the complexity,(e.g., the number of electronic gates), required to implement the exactapproach. Further, although for the sake of simplicity in discussion,embodiments discussed herein have assumed the 16-QAM constellation to benormalized to unit energy, it is expressly understood that othernormalizations are possible. In such embodiment(s), the √{square rootover (10)} factor would change, and the corresponding embodiments of theLLR calculation would accordingly change.

One example of an approximation that may be used to reduce thecomplexity of the LLR equations is through the max-log approximation.Equation (20) can be used to express the generic format of equations(12)-(19):

LLR=log [exp(A)+exp(B)]≈max[A,B]  (20)

which is valid in medium to high signal-to-noise ratio (SNR) cases.Using equation (20), expanding the matrices within equations (12)-(19),and substituting for T, the reduced complexity LLR equations can beexpressed as in FIGS. 4A-4H.

Simulations have shown that the high SNR max-log approximation resultsin only a loss of 0.1-0.2 dB when compared to the optimal (exact) LLRvalues, but at a much lower implementation complexity. The equations ofFIGS. 4A-4H are a preferred approach for obtaining soft information whenthe DCM mode is used. It is expressly understood that anytransformations or simplifications of these equations are within thespirit and scope of this disclosure and the appended claims.

Although an example of both an exact and approximate methods fordetermining the LLR are disclosed, other techniques may be used todetermine the LLR, and the present disclosure is not limited toparticular formulas. Moreover, other statistical methods to determinethe presence of errors, as known to one killed in the art, may be usedand are within the spirit and scope of the present disclosure.

It is expressly understood that occasionally, such as when narrowbandinterference is present in-band, the noise variance across tones is notuniform. In such an instance, the LLR equation for b_(4k) can be writtenas:

${{LLR}\left( b_{4k} \right)} = {\log\left\lbrack \frac{\sum\limits_{\substack{b_{{4k} + 1}b_{4{({k + p})}} \\ b_{{4{({k + p})}} + 1}}}{\Pr \left( {{\left. Y_{k} \middle| b_{4k} \right. = 0},b_{{4k} + 1},b_{4{({k + p})}},b_{{4{({k + p})}} + 1}} \right)}}{\sum\limits_{\substack{b_{{4k} + 1}b_{4{({k + p})}} \\ b_{{4{({k + p})}} + 1}}}{\Pr \left( {{\left. Y_{k} \middle| b_{4k} \right. = 1},b_{{4k} + 1},b_{4{({k + p})}},b_{{4{({k + p})}} + 1}} \right)}} \right\rbrack}$

$\begin{matrix}\begin{matrix}{= {\log\left\lbrack \frac{\sum\limits_{\substack{{{b_{{4k} + 1}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}}, \\ b_{4k} = 0}}{\exp \begin{bmatrix}{- \left( {\alpha_{k} - {{H_{k}}^{2}{TS}_{R}}} \right)^{T}} \\{W_{k}^{- 1}{H_{k}}^{- 2}} \\\left( {\alpha_{k} - {{H_{k}}^{2}{TS}_{R}}} \right)\end{bmatrix}}}{\sum\limits_{\substack{{{b_{{4k} + 1}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}}, \\ b_{4k} = 1}}{\exp \begin{bmatrix}{- \left( {\alpha_{k} - {{H_{k}}^{2}{TS}_{R}}} \right)^{T}} \\{W_{k}^{- 1}{H_{k}}^{- 2}} \\\left( {\alpha_{k} - {{H_{k}}^{2}{TS}_{R}}} \right)\end{bmatrix}}} \right\rbrack}} \\{where} \\{W_{k} = {\begin{bmatrix}\frac{\sigma_{k}^{2}}{2} & 0 \\0 & \frac{\sigma_{k + p}^{2}}{2}\end{bmatrix}.}}\end{matrix} & (21)\end{matrix}$

By expanding equation (21) and eliminating the terms that do not dependon S_(R), equation (21) can be re-written as:

$\begin{matrix}{{{LLR}\left( b_{4k} \right)} = {\log\left\lbrack \frac{\sum\limits_{\substack{{{b_{{4k} + 1}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}}, \\ b_{4k} = 0}}\; \left\lbrack {S_{R}^{T}{{TW}_{k}^{- 1}\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack}{\sum\limits_{\substack{{{b_{{4k} + 1}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}}, \\ b_{4k} = 1}}\; \left\lbrack {S_{R}^{T}{{TW}_{k}^{- 1}\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack} \right\rbrack}} & (22)\end{matrix}$

The LLR equations for b_(4k+1), b_(4k+2), b_(4k+3), b_(4(k+p)),b_(4(k+p)+1), b_(4(k+p)+2) and b_(4(k+p)+3) can also be rewritten insimilar fashion. It should be apparent that, in view of the teachings ofthe present disclosure, the corresponding equations for those bits are:

${{{LLR}\left( b_{{4k} + 1} \right)} = {\log\left\lbrack \frac{\sum\limits_{\substack{{{b_{4k}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}}, \\ b_{{4k} + 1} = 0}}{\exp \left\lbrack {S_{R}^{T}{{TW}_{k}^{- 1}\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack}}{\sum\limits_{\substack{{{b_{4k}b_{4{({k + p})}}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}}, \\ b_{{4k} + 1} = 1}}{\exp \left\lbrack {S_{R}^{T}{{TW}_{k}^{- 1}\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack}} \right\rbrack}},{{{LLR}\left( b_{4{({k + p})}} \right)} = {\log\left\lbrack \frac{\sum\limits_{\substack{{{b_{4k}b_{{4k} + 1}b_{{4{({k + p})}} + 1}} \in {\{{0,1}\}}}, \\ b_{4{({k + p})}} = 0}}{\exp \left\lbrack {S_{R}^{T}{{TW}_{k}^{- 1}\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack}}{\sum\limits_{\substack{b_{4k}b_{{4k} + 1}b_{{4{({k + p})}} + 1} \\ {\in {\{{0,1}\}}},{b_{4{({k + p})}} = 1}}}{\exp \left\lbrack {S_{R}^{T}{{TW}_{k}^{- 1}\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack}} \right\rbrack}},{{{LLR}\left( b_{{4{({k + p})}} + 1} \right)} = {\log\left\lbrack \frac{\sum\limits_{\substack{{{b_{4k}b_{{4k} + 1}b_{4{({k + p})}}} \in {\{{0,1}\}}}, \\ b_{{4{({k + p})}} + 1} = 0}}{\exp \left\lbrack {S_{R}^{T}{{TW}_{k}^{- 1}\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack}}{\sum\limits_{\substack{{{b_{4k}b_{{4k} + 1}b_{4{({k + p})}}} \in {\{{0,1}\}}}, \\ b_{{4{({k + p})}} + 1} = 1}}{\exp \left\lbrack {S_{R}^{T}{{TW}_{k}^{- 1}\left( {{2\alpha_{k}} - {{H_{k}}^{2}{TS}_{R}}} \right)}} \right\rbrack}} \right\rbrack}},{{{LLR}\left( b_{{4k} + 2} \right)} = {\log\left\lbrack \frac{\sum\limits_{\substack{{{b_{{4k} + 3}b_{{4{({k + p})}} + 2}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}}, \\ b_{{4k} + 2} = 0}}{\exp \left\lbrack {S_{I}^{T}{{TW}_{k}^{- 1}\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}}{\sum\limits_{\substack{{{b_{{4k} + 3}b_{{4{({k + p})}} + 2}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}}, \\ b_{{4k} + 2} = 1}}{\exp \left\lbrack {S_{I}^{T}{{TW}_{k}^{- 1}\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}} \right\rbrack}},{{{LLR}\left( b_{{4k} + 3} \right)} = {\log\left\lbrack \frac{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4{({k + p})}} + 2}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}}, \\ b_{{4k} + 3} = 0}}{\exp \left\lbrack {S_{I}^{T}{{TW}_{k}^{- 1}\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}}{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4{({k + p})}} + 2}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}}, \\ b_{{4k} + 3} = 1}}{\exp \left\lbrack {S_{I}^{T}{{TW}_{k}^{- 1}\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}} \right\rbrack}},{{{LLR}\left( b_{{4{({k + p})}} + 2} \right)} = {\log\left\lbrack \frac{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4k} + 3}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}}, \\ b_{{4{({k + p})}} + 2} = 0}}{\exp \left\lbrack {S_{I}^{T}{{TW}_{k}^{- 1}\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}}{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4k} + 3}b_{{4{({k + p})}} + 3}} \in {\{{0,1}\}}}, \\ b_{{4{({k + p})}} + 2} = 1}}{\exp \left\lbrack {S_{I}^{T}{{TW}_{k}^{- 1}\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}} \right\rbrack}},{{{LLR}\left( b_{{4{({k + p})}} + 3} \right)} = {{\log \left\lbrack \frac{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4k} + 3}b_{{4{({k + p})}} + 2}} \in {\{{0,1}\}}}, \\ b_{{4{({k + p})}} + 3} = 0}}{\exp \left\lbrack {S_{I}^{T}{{TW}_{k}^{- 1}\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}}{\sum\limits_{\substack{{{b_{{4k} + 2}b_{{4k} + 3}b_{{4{({k + p})}} + 2}} \in {\{{0,1}\}}}, \\ b_{{4{({k + p})}} + 3} = 1}}{\exp \left\lbrack {S_{I}^{T}{{TW}_{k}^{- 1}\left( {{2\beta_{k}} - {{H_{k}}^{2}{TS}_{I}}} \right)}} \right\rbrack}} \right\rbrack}.}}$

Expanding the matrices within equation (22), and substituting for T, thereduced complexity LLR equations can be expressed as in FIGS. 4I-4P.

FIG. 5 is a flowchart of an exemplary embodiment of DCM decoding. Inthis example embodiment, separate tones distanced by some predeterminedspacing are grouped into a vector (block 510). The vector is passedthrough FFT 40 to produce symbol₃ and symbol₄ (block 510). Bits b₀, b₁,b₂, b₃, b₄, b₅, b₆, b₇ are jointly recovered from third and fourthinputs using exact Max-Log equations (block 530). It should beunderstood that the references to b₀, b₁, b₂, b₃, b₄, b₅, b₆, b₇ hereinare generic references to exemplary bits and should in no fashion beconsidered a limitation.

FIG. 6 is a flowchart of an exemplary embodiment of decoding usingapproximation. Blocks 610 and 620 correspond to blocks 510 and 520 ofFIG. 5. However, with this embodiment, bits b₀, b₁, b₂, b₃, b₄, b₅, b₆,b₇ are jointly recovered from third and fourth inputs using a Max-Logapproximation (block 630).

The systems and methods described above may be implemented on anygeneral-purpose computer with sufficient processing power, memoryresources, and network throughput capability to handle the necessaryworkload placed upon it. FIG. 7 illustrates an exemplary,general-purpose computer system suitable for implementing one or moreembodiments of a system to respond to signals as disclosed herein.Computer system 70 includes processor 72 (which may be referred to as acentral processor unit or CPU) that is in communication with memorydevices including secondary storage 74, read only memory (ROM) 76,random access memory (RAM) 78, input/output (I/O) 75 devices, and host77. The processor may be implemented as one or more CPU chips.

Secondary storage 74 typically comprises one or more disk drives or tapedrives and is used for non-volatile storage of data and as an over-flowdata storage device if RAM 78 is not large enough to hold all workingdata. Secondary storage 74 may be used to store programs that are loadedinto RAM 78 when such programs are selected for execution. ROM 76 is anon-volatile memory device that typically has a small memory capacityrelative to the larger memory capacity of secondary storage. RAM 78 isused to store volatile data and perhaps to store instructions. Access toboth ROM 76 and RAM 78 is typically faster than to secondary storage 74.

I/O devices 75 may include printers, video monitors, liquid crystaldisplays (LCDs), touch screen displays, keyboards, keypads, switches,dials, mice, track balls, voice recognizers, card readers, paper tapereaders, or other well-known input devices. Host 77 may interface toEthernet cards, universal serial bus (USB), token ring cards, fiberdistributed data interface (FDDI) cards, wireless local area network(WLAN) cards, and other well-known network devices. Host 77 may enableprocessor 72 to communicate with an Internet or one or more intranets.With such a network connection, it is contemplated that processor 72might receive information from the network, or might output informationto the network in the course of performing the above-describedprocesses.

Processor 72 executes instructions, codes, computer programs, andscripts which it accesses from hard disk, floppy disk, optical disk(these various disk based systems may all be considered secondarystorage 74), ROM 76, RAM 78, or the host 77.

The systems and methods described above may be implemented on deviceswith a MAC and a PHY. FIG. 8 illustrates an exemplary system 80containing a MAC 82, a MAC-PHY interface 84, and a PHY 86. MAC 82 iscapable, in this embodiment, of communicating with PHY 86 throughMAC-PHY interface 84. MAC-PHY interface 84 may be a controller,processor, direct electrical connection, or any other system or method,logical or otherwise, that facilitates communication between MAC 82 andPHY 86. It is expressly understood that MAC 82, MAC-PHY interface 84,and PHY 86 may be implemented on a single electrical device, such as anintegrated controller, or through the use of multiple electricaldevices. It is further contemplated that MAC 82, MAC-PHY interface 84,and PHY 86 may be implemented through firmware on an embedded processor,or otherwise through software on a general purpose CPU, or may beimplemented as hardware through the use of dedicated components, or acombination of the above choices. Any implementation of a deviceconsistent with this disclosure containing a MAC and a PHY may contain aMAC-PHY interface. It is therefore expressly contemplated that thedisclosed systems and methods may be used with any device with a MAC anda PHY.

Many modifications and other embodiments of the invention will come tomind to one skilled in the art to which this invention pertains havingthe benefit of the teachings presented in the foregoing descriptions,and the associated drawings. Therefore, it is to be understood that theinvention is not to be limited to the specific embodiments disclosed.Although specific terms are employed herein, they are used in a genericand descriptive sense only and not for purposes of limitation.

1. A communications system, comprising: a dual carrier modulation (DCM)encoder for applying a pre-transmission function to at least one 16-QAMinput symbol and mapping resulting transformed symbols onto at least onelarger constellation prior to transmission.
 2. The system of claim 1,wherein the pre-transmission function is a mixing matrix, T, such that$T = {{\frac{1}{\sqrt{17}}\begin{bmatrix}4 & 1 \\1 & {- 4}\end{bmatrix}}.}$
 3. The system of claim 1, wherein the pre-transmissionfunction is a mixing matrix, T, such that$T = {{\frac{1}{\sqrt{17}}\begin{bmatrix}{\pm 4} & {\pm 1} \\{\pm 1} & {\pm 4}\end{bmatrix}}.}$
 4. The system of claim 3, wherein the signs within themixing matrix are selected to ensure unique mapping of two 16-QAM inputsymbols onto at least one larger constellation.
 5. The system of claim1, wherein the at least one larger constellation is a 256-pointconstellation.
 6. The system of claim 1, wherein two 16-QAM inputsymbols are mapped onto a single larger constellation.
 7. The system ofclaim 6, wherein the two 16-QAM input symbols are mapped onto a seconddistinct single larger constellation.
 8. The system of claim 1, whereineight (8) bits corresponding to two 16-QAM input symbols are directlymapped onto a single larger constellation.
 9. The system of claim 8,wherein the eight (8) bits corresponding to two 16-QAM input symbols aredirectly mapped onto a second distinct single larger constellation. 10.The system of claim 1, wherein the DCM encoder maps the resultingtransformed symbols onto tones of an inverse fast-Fourier transform(IFFT) with a pre-determined frequency separation between the tones. 11.The system of claim 10, wherein the pre-determined separation betweenthe tones is an integer between one (1) and one hundred (100).
 12. Thesystem of claim 10, wherein the pre-determined separation between thetones is fifty (50).
 13. The system of claim 1, wherein the system is anultra-wideband system.
 14. The system of claim 1, wherein the at leastone 16-QAM input symbol is Gray-coded.
 15. The system of claim 1,wherein the pre-transmission function is orthonormal.
 16. Acommunications system, comprising: a dual carrier modulation (DCM)encoder for applying a pre-transmission function and mapping resultingtransformed symbols onto at least one larger constellation prior totransmission, wherein an input to the dual carrier modulation (DCM)encoder is a first symbol selected from a 16-QAM constellation bearing afirst plurality of data elements and a second symbol selected from a16-QAM constellation bearing a second plurality data elements, the dualcarrier modulation (DCM) encoder performs a transform on the firstsymbol together with the second symbol to create first transformedsymbol and second transformed symbol.
 17. The system of claim 16,wherein each of said transformed symbols bear the data from all eightdata elements corresponding to said transformed symbols.
 18. The systemof claim 16, wherein each of said data elements comprising a single bitof data.
 19. A communications system, comprising: a dual carriermodulation (DCM) encoder for applying a pre-transmission function to atleast one 16-QAM input symbol and mapping resulting transformed symbolsonto at least one larger constellation prior to transmission, saidmapping comprising: mapping a first 4 bits onto a first input 16-QAMconstellation to produce a first symbol; mapping a second 4 bits onto asecond input 16-QAM constellation to produce a second symbol; groupingsaid first symbol and said second symbol into a vector; passing throughan orthonormal matrix producing a first transformed symbol and a secondtransformed symbol which are symbols belonging to a first and a secondconstellation.
 20. The system of claim 19, further comprising: mappingsaid first transformed symbol onto a first tone; and mapping said secondtransformed symbol onto a second tone.